From studiosus' answer to
A 3-manifold with fundamental group isomorphic to a surface group.
a closed 3-manifold cannot have a fundamental group isomorphic to that of a closed surface of genus $\geq 2$.
What about genus 1? I wonder if there exist a closed 3-manifold with fundamental group isomorphic to the fundamental group of the 2-dimensional torus or the Klein bottle.
Thanks in advance